monomial

A monomial is a polynomial consisting of a single term. In multivariate algebra, a monomial in variables x1, ..., xn over a ring R has the form c x1^a1 x2^a2 ... xn^an, where c is a nonzero element of R and each ai is a nonnegative integer. Exponents equal to zero effectively remove the corresponding variable from the term.

The total degree of a monomial is the sum a1 + a2 + ... + an. Monomials form a natural basis for polynomial rings over R when polynomials are viewed as finite sums of terms; any polynomial is a finite sum of monomials.

Examples include 3x^2 y^3, -7xy, and 5. Under the common convention that the coefficient must be nonzero, the zero polynomial is not a monomial. When multiplying monomials, coefficients multiply and exponents add: (c1 x^a y^b)(c2 x^m y^n) = (c1 c2) x^{a+m} y^{b+n}.

Applications and related concepts: monomials are the basic building blocks of polynomial rings. Monomial ideals are ideals generated by monomials and play a key role in computational algebra and algebraic geometry. Monomial orders, used to compare monomials, underpin algorithms such as Gröbner bases. Monomials also appear in combinatorics and invariant theory as simple yet fundamental objects in polynomial expressions.